Properties

Label 135240.e3
Conductor $135240$
Discriminant $8.363\times 10^{19}$
j-invariant \( \frac{76343005935514084}{694180580625} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -4364936, 3483833340])
 
gp: E = ellinit([0, -1, 0, -4364936, 3483833340])
 
magma: E := EllipticCurve([0, -1, 0, -4364936, 3483833340]);
 

\(y^2=x^3-x^2-4364936x+3483833340\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z \times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(4318, 255200\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $5.6955993212786954366226516355$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(1118, 0\right) \), \( \left(1293, 0\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-2410, 0\right) \), \( \left(1118, 0\right) \), \( \left(1293, 0\right) \), \((4318,\pm 255200)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 135240 \)  =  $2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $83629722757069440000 $  =  $2^{10} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} \cdot 23^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{76343005935514084}{694180580625} \)  =  $2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{-2} \cdot 13^{3} \cdot 23^{-4} \cdot 61^{3} \cdot 337^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.6450368064009394746713950996\dots$
Stable Faltings height: $1.0944590814066617309376919600\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $5.6955993212786954366226516355\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.19295979106366704940018696892\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 64 $  = $ 2\cdot2\cdot2\cdot2^{2}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L'(E,1) $ ≈ $ 4.3960866200652037102195307161382827050 $

Modular invariants

Modular form 135240.2.a.e

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{3} - q^{5} + q^{9} - 4q^{11} + 2q^{13} + q^{15} - 2q^{17} + 4q^{19} + O(q^{20}) \)  Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 5505024
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $4$ $I_2^{*}$ Additive -1 2 8 2
$23$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.24.0.19

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add nonsplit nonsplit add ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 3 1 - 1 1 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) - 0 0 - 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 135240.e consists of 3 curves linked by isogenies of degrees dividing 8.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{7}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-7})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-7})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.157351936.1 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.8.107503718560000.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.